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A105931
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a(1) = 1 then a(n) = a(n-1) - (-1)^ceiling(n/2)*a(floor(n/2)).
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1
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1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, 2, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2, 1, -1, 1, 2
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OFFSET
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1,2
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COMMENTS
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The asymptotic density of the occurrences of -1, 1, and 2 are 1/6, 1/2, and 1/3, respectively. - Amiram Eldar, Nov 30 2022
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LINKS
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FORMULA
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Multiplicative with a(2^e) = 2 if e is odd and -1 if e is even, and a(p^e) = 1 for p > 2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 1. (End)
Dirichlet g.f.: zeta(s)*(4^s+2^(s+1)-2)/(4^s+2^s). - Amiram Eldar, Jan 04 2023
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MATHEMATICA
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a[n_] := Module[{e = IntegerExponent[n, 2]}, If[e > 0, If[Mod[e, 2] == 1, 2, -1], 1]]; Array[a, 100] (* Amiram Eldar, Nov 30 2022 after the second PARI code *)
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PROG
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(PARI) a(n)=if(n<2, 1, a(n-1)-(-1)^ceil(n/2)*a(floor(n/2)))
(PARI) a(n)={my(e=valuation(n, 2)); if(e>0, if(e%2, 2, -1), 1)} \\ Andrew Howroyd, Aug 06 2018
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CROSSREFS
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KEYWORD
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sign,mult
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AUTHOR
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STATUS
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approved
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